Hardy & Wright (An Introduction to the Theory of Numbers) is written to minimize the required math background. H&W isn't really 'algebraic' or 'analytic'. There's no mention of homomorphism, groups, or maps. Complex numbers appear, but mostly in algebraic number fields. Ideals of number fields are sampled briefly, discussing how factorization changes. Complex variables do not appear. The zeta function is used, but only with real argument >1. No mention of the zeta functional equation or Riemann Hypothesis. Power series are used extensively in the Partitions chapter, but only as a combinatorial device. They do teach O-notation, and use it extensively. Use of calculus is postponed to the later chapters. The proof of the Prime Number Theorem mostly follows Selberg's elementary proof. The preface mentions that they omit quadratic forms, and are disjoint from Dickson's IttToN (same title, but about Diophantine Equations). I think it's a great book; it's been well worth the $12, even in 1960 $. It's not really aimed at youngsters. I've considered the project of annotating it, to explain the parts that confused me. But it's hard to recreate my younger viewpoint. When I first got H&W, there were entire chapters that made little or no sense -- I could read the theorems, but the proofs were too hard, and often the point of the theorem was a puzzle. I went back many times, and the dark areas have receded. Is it worth doing a math book that makes sense for bright kids, and tries to explain everything? Or is it better to leave some mysteries? Rich ----------- Quoting Gareth McCaughan <gareth.mccaughan@pobox.com>:
On 22/04/2015 22:13, Dan Asimov wrote:
G.H. Hardy & E.M. Wright, An Introduction to the Theory of Numbers algebraic but not analytic
I'm not sure that's a very accurate description. E.g., on the analytic side they get as far as proving the Prime Number Theorem (but no further; e.g., no Dirichlet) and on the algebraic side they get as far as discussing the primes in quadratic fields with unique factorization (but no further; e.g., no ideal class group). It's always felt to me like quite an "analytic" book, which makes sense given Hardy's work.
But I'm not a number theorist; perhaps my idea of what counts as algebraic or analytic is distorted somehow.
-- g
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